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Let $\{x_i\}_{i=1}^n$ be a sample from a multivariate Gaussian distribution ${\cal N}(0, \Sigma_X)$ and $\{y_i\}_{i=1}^m$ be a sample from ${\cal N}(0, \Sigma_Y)$.

Are there hypothesis tests for $\Sigma_X = \Sigma_Y$? Pointers to relevant literature would be very appreciated.

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The Mauchly's test allows to test if a given covariance matrix is proportional to a reference (identity or other) and is available through mauchly.test() under R. It is mostly used in repeated-measures design (to test (1) if the dependent variable VC matrices are equal or homogeneous, and (2) whether the correlations between the levels of the within-subjects variable are comparable--altogether, this is known as the sphericity assumption).

Box’s M statistic is used (in MANOVA or LDA) to test for homogeneity of covariance matrices, but as it is very sensitive to normality it will often reject the null (R code not available in standard packages).

Covariance structure models as found in Structural Equation Modeling are also an option for more complex stuff (although in multigroup analysis testing for the equality of covariances makes little sense if the variances are not equal), but I have no references to offer actually.

I guess any textbook on multivariate data analysis would have additional details on these procedures. I also found this article for the case where normality assumption is not met:

Aslam, S and Rocke, DM. A robust testing procedure for the equality of covariance matrices, Computational Statistics & Data Analysis 49 (2005) 863-874

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